Weyl asymptotics for non-self-adjoint operators with small multiplicative random perturbations

Johannes Sjöstrand[1]

  • [1] Centre de Mathématiques Laurent Schwartz, UMR 7640, CNRS and École polytechnique, F-91128 Palaiseau Cedex

Séminaire Équations aux dérivées partielles (2007-2008)

  • Volume: 2007-2008, page 1-16

Abstract

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We study the Weyl asymptotics of the distribution of eigenvalues of non-self-adjoint (pseudo)differential operators with small random multiplicative perturbations in arbitrary dimension. We were led to quite essential improvements of many of the probabilistic aspects.

How to cite

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Sjöstrand, Johannes. "Weyl asymptotics for non-self-adjoint operators with small multiplicative random perturbations." Séminaire Équations aux dérivées partielles 2007-2008 (2007-2008): 1-16. <http://eudml.org/doc/11176>.

@article{Sjöstrand2007-2008,
abstract = {We study the Weyl asymptotics of the distribution of eigenvalues of non-self-adjoint (pseudo)differential operators with small random multiplicative perturbations in arbitrary dimension. We were led to quite essential improvements of many of the probabilistic aspects.},
affiliation = {Centre de Mathématiques Laurent Schwartz, UMR 7640, CNRS and École polytechnique, F-91128 Palaiseau Cedex},
author = {Sjöstrand, Johannes},
journal = {Séminaire Équations aux dérivées partielles},
keywords = {semi-classical pseudo-differential operators; non-self-adjoint operators; random perturbations},
language = {eng},
pages = {1-16},
publisher = {Centre de mathématiques Laurent Schwartz, École polytechnique},
title = {Weyl asymptotics for non-self-adjoint operators with small multiplicative random perturbations},
url = {http://eudml.org/doc/11176},
volume = {2007-2008},
year = {2007-2008},
}

TY - JOUR
AU - Sjöstrand, Johannes
TI - Weyl asymptotics for non-self-adjoint operators with small multiplicative random perturbations
JO - Séminaire Équations aux dérivées partielles
PY - 2007-2008
PB - Centre de mathématiques Laurent Schwartz, École polytechnique
VL - 2007-2008
SP - 1
EP - 16
AB - We study the Weyl asymptotics of the distribution of eigenvalues of non-self-adjoint (pseudo)differential operators with small random multiplicative perturbations in arbitrary dimension. We were led to quite essential improvements of many of the probabilistic aspects.
LA - eng
KW - semi-classical pseudo-differential operators; non-self-adjoint operators; random perturbations
UR - http://eudml.org/doc/11176
ER -

References

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  1. W. Bordeaux-Montrieux, thèse, en préparation 
  2. E.B. Davies, Semi-classical states for non-self-adjoint Schrödinger operators, Comm. Math. Phys. 200(1999), 35–41. Zbl0921.47060MR1671904
  3. N.Dencker, J.Sjöstrand, M.Zworski, Pseudospectra of semiclassical (pseudo-) differential operators, Comm. Pure Appl. Math., 57(3)(2004), 384–415. Zbl1054.35035MR2020109
  4. M. Dimassi, J. Sjöstrand, Spectral asymptotics in the semi-classical limit, London Math. Soc. Lecture Notes Ser., 268, Cambridge Univ. Press, (1999). Zbl0926.35002MR1735654
  5. I.C. Gohberg, M.G. Krein, Introduction to the theory of linear non-selfadjoint operators, Translations of mathematical monographs, Vol 18, AMS, Providence, R.I. (1969). Zbl0181.13504MR246142
  6. M. Hager, Instabilité spectrale semiclassique pour des opérateurs non-autoadjoints. I. Un modèle, Ann. Fac. Sci. Toulouse Math. (6)15(2)(2006), 243–280. Zbl1131.34057MR2244217
  7. M. Hager, Instabilité spectrale semiclassique d’opérateurs non-autoadjoints. II. Ann. Henri Poincaré, 7(6)(2006), 1035–1064. Zbl1115.81032
  8. M. Hager, J. Sjöstrand, Eigenvalue asymptotics for randomly perturbed non-selfadjoint operators, Math. Annalen, to appear, http://arxiv.org/abs/math/0601381, Zbl1151.35063
  9. A. Melin, J. Sjöstrand, Determinants of pseudodifferential operators and complex deformations of phase space, Methods and Applications of Analysis, 9(2)(2002), 177-238. http://xxx.lanl.gov/abs/math.SP/0111292 Zbl1082.35176MR1957486
  10. K. Pravda Starov Etude du pseudo-spectre d’opérateurs non auto-adjoints, thèse Rennes 2006, http://tel.archives-ouvertes.fr/tel-00109895 
  11. R.T. Seeley, Complex powers of an elliptic operator. 1967 Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966) pp. 288–307, Amer. Math. Soc., Providence, R.I. Zbl0159.15504MR237943
  12. J. Sjöstrand, Resonances for bottles and trace formulae, Math. Nachr., 221(2001), 95–149. Zbl0979.35109MR1806367
  13. J. Sjöstrand, Eigenvalue distribution for non-self-adjoint operators with small multiplicative random perturbations, http://arxiv.org/abs/0802.3584 . Zbl1194.47058
  14. J. Sjöstrand, M. Zworski, Elementary linear algebra for advanced spectral problems, Ann. Inst. Fourier, 57(7)(2007), 2095-2141. Zbl1140.15009
  15. M. Zworski, A remark on a paper of E.B. Davies, Proc. A.M.S. 129(2001), 2955–2957. Zbl0981.35107MR1840099

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